LABORATORY-DERIVED FRICTION LAWS AND THEIR APPLICATION TO SEISMIC FAULTING

Transcription

1 Annu. Rev. Earth Planet. Sci : Copyright c 1998 by Annual Reviews. All rights reserved LABORATORY-DERIVED FRICTION LAWS AND THEIR APPLICATION TO SEISMIC FAULTING Chris Marone Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; cjm@westerly.mit.edu KEY WORDS: earthquake faults, frictional properties and constitutive laws, physics of friction, granular fault gouge, earthquake afterslip ABSTRACT This paper reviews rock friction and the frictional properties of earthquake faults. The basis for rate- and state-dependent friction laws is reviewed. The friction state variable is discussed, including its interpretation as a measure of average asperity contact time and porosity within granular fault gouge. Data are summarized showing that friction evolves even during truly stationary contact, and the connection between modern friction laws and the concept of static friction is discussed. Measurements of frictional healing, as evidenced by increasing static friction during quasistationary contact, are reviewed, as are their implications for fault healing. Shear localization in fault gouge is discussed, and the relationship between microstructures and friction is reviewed. These data indicate differences in the behavior of bare rock surfaces as compared to shear within granular fault gouge that can be attributed to dilation within fault gouge. Physical models for the characteristic friction distance are discussed and related to the problem of scaling this parameter to seismic faults. Earthquake afterslip, its relation to laboratory friction data, and the inverse correlation between afterslip and shallow coseismic slip are discussed in the context of a model for afterslip. Recent observations of the absence of afterslip are predicted by the model. INTRODUCTION Since their introduction nearly 20 years ago, friction constitutive laws of the slip rate and state variable type (Dieterich 1979, Ruina 1983) have emerged as /98/ $

2 644 MARONE powerful tools for investigating the mechanics of earthquakes and faulting. The incorporation of a state variable provided a means of describing complex friction memory effects and history dependence, and the resulting constitutive laws have been extremely successful in modeling laboratory data. These laws are capable of reproducing virtually the entire range of observed seismic and interseismic fault behaviors, ranging from preseismic slip and earthquake nucleation (Dieterich 1986, 1992, Stuart & Tullis 1995, Roy & Marone 1996, Tullis 1996, Dieterich & Kilgore 1996a) to coseismic rupture (Tse & Rice 1986, Okubo 1989, Cochard & Madariaga 1994, Ben-Zion & Rice 1995, 1997, Boatwright & Cocco 1996) and earthquake afterslip (Marone et al 1990, Wennerberg & Sharp 1997). In addition, the laws have been widely used to describe systematic variations in seismic behavior, including the depth of seismic faulting (Tse & Rice 1986, Marone & Scholz 1988, Blanpied et al 1991), variation of stress drop with earthquake recurrence interval (Scholz et al 1986, Kanamori & Allen 1986, Vidale et al 1994, Marone et al 1995), seismic slip complexity (Takashi 1992, Rice 1993, Rice & Ben-Zion 1996), variations in the stability and seismic coupling at subduction zones (Scholz 1990, Scholz & Campos 1995), and characteristics of aftershock rate decay (Dieterich 1994, Gross & Kisslinger 1997). The past few years have seen continued growth in the application and physical understanding of friction constitutive laws. In modeling applications, improvements in the quality and spatial resolution of seismic and geodetic observations have led to significant advances in the evaluation of laboratory-based models. In the laboratory, a consensus has emerged concerning several aspects of friction data for rock and granular materials (crushed rock, sand, or powders) used to simulate fault gouge. Such studies have resolved, under a limited range of conditions, issues including the nature of frictional state evolution, the role of dilatancy and fault gouge in friction velocity dependence, and the effect of shear strain and displacement on scaling parameters and friction behavior. In addition, discrepancies between results obtained in different testing configurations are now more clearly understood. However, in spite of their utility and widespread use, laboratory-based friction laws and their application in nature have a number of shortcomings. Chief among these are perhaps the empirical nature of the laws and the scaling problem associated with extrapolating results outside of the laboratory range of conditions. Thus, in this review, I have chosen to focus on recent laboratory results and field observations related to the scaling problem and on modeling studies aimed at applying laboratory-based laws to seismic faulting. To this end, I summarize recent results related to the rate of frictional healing, work on the effects of displacement and strain on frictional behavior, and studies of postseismic slip that can be used to infer the rheology of mature faults. For the

3 FRICTION LAWS AND FAULTING 645 most part, I do not revisit topics covered in the excellent summaries of earlier reviews on this subject (Rudnicki 1980, 1988, Mavko 1981, Sibson 1986, Tullis 1988, Scholz 1989, 1990, Kanamori 1994). Yet, even with these restrictions, the topic is exceedingly broad, drawing from the detailed work of laboratory, modeling, theoretical, and observational studies. Thus, I attempt only a brief summary of progress in selected areas. Unfortunately, this rather limited scope does not encompass several fruitful lines of study, and I can only suggest a few initiation points for readers interested in works on spatio-temporal complexity and frequency-magnitude scaling of seismicity (Horowitz & Ruina 1989, Shaw et al 1992, Abercrombie & Leary 1993, Rice 1993, Wesnousky 1994, Shaw 1994, Ben-Zion & Rice 1995, Ben-Zion 1996, Sornette et al 1996, Heimpel 1996, 1997); fluid, poro-elastic, and thermal effects on friction and the seismic cycle (Mase & Smith 1987, Blanpied et al 1991, 1992, 1997, Chester & Higgs 1992, Sleep & Blanpied 1992, 1994, Chester 1994, 1995, Sleep 1994, 1995a,b, 1997, Hickman et al 1995, Segall & Rice 1995, Shaw 1995, Miller 1996, Miller et al 1996, Karner et al 1997); elasto-dynamic rupture propagation, including studies of opening mode waves and Heaton pulses (Cochard & Madariaga 1994, 1996, Perrin et al 1995, Beeler & Tullis 1996, Andrews & Ben-Zion 1997); dynamical models of earthquakes employing idealized friction laws and their connection to continuum-based models (Carlson & Langer 1989, Huang & Turcotte 1992, Carlson et al 1991, Rice 1993, Shaw 1995, Rice & Ben-Zion 1996, Schmittbuhl et al 1996, Main 1996, Rundle et al 1996); studies of granular materials and acoustic fluidization related to fault mechanics and rupture propagation (Melosh 1979, 1996, Lorig & Hobbs 1990, Mora & Place 1994, Scott 1996); studies of rupture nucleation using laboratory-derived friction laws (Dieterich 1986, 1992, 1994, Yamashita & Ohnaka 1991, Shibazaki & Matsu ura 1992, 1995, Roy & Marone 1996, Kato & Hirasawa 1996); and laboratory and theoretical works focused on building detailed physical models of base friction using contact theory (Yoshioka & Scholz 1989a,b, Stesky & Hannan 1989, Biegel et al 1992, Boitnott et al 1992, Wang & Scholz 1994, 1995). I begin with a brief historical introduction to the friction laws, which provides a background for understanding the significance of some important recent results concerning the nature of frictional state evolution and frictional healing. These works illuminate the relationship between so-called static and dynamic friction, terms that are dated but still of practical use, and provide a connection between laboratory observations of frictional healing and seismic estimates of the rate of fault healing. In the subsequent section, I consider the effects of fault gouge and strain on friction behavior and constitutive parameters and their implications for the scaling problem. In this case, the effects of shear localization, dilation, and net displacement are quite important. These results

4 646 MARONE are closely related to the mechanistic interpretation of the critical slip distance (defined below) and the problem of scaling this parameter to seismic faulting. A new mechanical interpretation of the critical slip distance for fault gouge is discussed in that section. Finally, application of the laboratory-derived friction laws to earthquake afterslip and the rheology of mature fault zones is reviewed. In this case, seismic data and field observations are of sufficient quality to provide constraints on laboratory-based friction laws and associated models. Because the vast majority of laboratory experiments have been carried out at room temperature and with quartzo-feldspathic materials, I focus primarily on these experiments. FUNDAMENTALS OF LABORATORY-DERIVED FRICTION LAWS Slip rate and state variable constitutive laws for rock friction were first introduced by Dieterich (1979, 1981), Ruina (1983), and Rice (Rice 1983, Rice & Ruina 1983). The laws were the outgrowth of a broad effort to understand rock friction, beginning in its modern form with the work of Brace and coworkers (Brace & Byerlee 1966, 1970, Byerlee 1967) and including several seminal works in the succeeding decade (Dieterich 1972, 1978, Scholz et al 1972, Ohnaka 1973, Scholz & Engelder 1976, Byerlee 1978, Logan 1978, Stesky 1978, Teufel & Logan 1978). These early works were designed to study frictional instability as a possible mechanism for repetitive stick-slip failure and the seismic cycle. The works made two primary contributions of direct relevance here. The first involved the recognition of frictional stability as a system response determined by the contacting surfaces and their elastic surroundings (e.g. Cook 1981). This led to a fundamental shift in the way laboratory studies were carried out. Simple mapping studies of the stability boundary between stick-slip and stable sliding were recognized as having limited value, and they were gradually replaced by more detailed studies in which friction data were subject to sophisticated modeling in order to separate stability and friction properties from apparatus effects (see Tullis 1988 for a summary). As a result, friction data were increasingly cast in terms of constitutive parameters and constitutive laws that could be readily applied in a variety of mechanical settings, including those of seismogenic faults (e.g. Dieterich 1979, 1981, Dieterich & Conrad 1984, Okubo & Dieterich 1984, Weeks & Tullis 1985, Blanpied & Tullis 1986, Lockner et al 1986, Ohnaka 1986, Shimamoto & Logan 1986, Tullis & Weeks 1986, Olsson 1988, Biegel et al 1989, Sammis & Biegel 1989, Marone et al 1990, Wong & Zhao 1990, Blanpied et al 1991, 1995, Reinen et al 1991, 1992, 1994, Chester & Higgs 1992, Linker & Dieterich 1992, Marone et al 1992, Steacy & Sammis 1992, Wong et al 1992, Kilgore et al 1993, Marone &

5 FRICTION LAWS AND FAULTING 647 Kilgore 1993, Reinen & Weeks 1993, Beeler et al 1994, 1996, Chester 1994, 1995, Dieterich & Kilgore 1994, 1996a,b, Gu & Wong 1994, Kato et al 1994, Marone & Cox 1994, Sammis & Steacy 1994, Scott et al 1994, Wang & Scholz 1994, Beeler & Tullis 1997, Blanpied et al 1997, Karner et al 1997). The use of quantitative friction constitutive laws relating stress and strain or displacement also provided a context within which friction data and laboratory-based continuum models could be subject to rigorous stability analysis (Rice & Ruina 1983, Gu et al 1984, Blanpied & Tullis 1986, Horowitz 1988, Dieterich & Linker 1992). The second contribution involved detailed measurements of the velocity dependence of sliding friction and the time dependence of static friction (Figure 1) (Dieterich 1972, 1978, 1979, 1981, Scholz et al 1972, Engelder & Scholz 1976, Scholz & Engelder 1976, Teufel & Logan 1978, Johnson 1981). These data could not be rationalized within the context of existing friction laws and required a new framework for understanding rock friction. Two aspects of these data are particularly relevant here, and I briefly summarize the original results, with recent data added where appropriate. Figure 1a shows static friction measurements for granite and simulated fault gouge. The data are obtained from experiments (Figure 1b) in which loading and steady frictional sliding are interrupted for a specified time, here for 10 and 100 s, after which loading resumes at the original rate, a so-called slidehold-slide test. Although the term static friction implies a measure of strength in the absence of slip, this is in fact not the case (e.g. Scholz et al 1972). Static friction µ s is defined (Dieterich 1972) as the maximum value following a hold period, and this corresponds to the point at which slip velocity first reaches the pre-hold value. Because static friction increases with hold time, measurements of it must be carried out by first sliding and then holding so that the initiation time of the hold is known. Dieterich (1972) showed that static friction increases logarithmically with hold time, and subsequent results indicate that the rate is somewhat higher for rock than for simulated fault gouge (Figure 1a). These data could be fit by empirical, time-dependent strengthening laws and were consistent with creep and contact indentation models in use at that time; however, only the static friction values µ s are fit by such laws. The laws are not capable of describing details of the time- and displacement-dependent changes in friction that accompany changes in static friction (Figure 1b), nor are they able to explain static friction in the context of models for velocitydependent dynamic friction. At the same time, measurements of dynamic friction for rock and gouge (Figure 1c) showed that sliding friction decreases with velocity, a phenomenon known as velocity weakening (Scholz & Engelder 1976). Early work showed that bare rock surfaces exhibited velocity-weakening friction over a range of

6 648 MARONE

7 FRICTION LAWS AND FAULTING 649 velocities. Subsequent work indicated velocity strengthening for pervasive shear within granular gouge, with a transition to velocity weakening for localized shear, as discussed more fully below; however, for both rock and fault gouge, that data indicate that friction evolves over a finite slip distance upon a sudden change in loading velocity (Figure 1d ). The measurements show that dynamic friction exhibits velocity weakening (i.e. µ d decreases with increasing slip velocity) over a broad range of velocities, extending up to centimeters per second in the case of fault gouge (Figure 1c) and meters per second for shear of gabbro (Tsutsumi & Shimamoto 1997). In some cases at the higher velocities, friction of rock exhibits a transition to velocity strengthening (Blanpied et al 1987, Kilgore et al 1993, Weeks 1993), which is presumed to arise from the effects of frictional heating; however, such effects apparently vary with experimental configuration and other factors that remain poorly understood (Kilgore et al 1993, Spray 1993, Tsutsumi & Shimamoto 1997, Blanpied et al 1998). A major problem posed by these observations was that of how static and dynamic friction measurements could be related. In particular, although the observations seem consistent, if hold time for static friction measurements is taken proportional to inverse average velocity, the data indicate that friction is not a single-valued function of velocity (Figure 1d ). Thus, a simple velocitydependent friction law is not sufficient, a point that continues to be at the root of differences between physically based and dynamical models of friction (Shaw et al 1992, Rice 1993, Myers et al 1996, Rice & Ben-Zion 1996). The other type of friction law in common use at that time, the slip-weakening law favored by those modeling dynamic shear rupture (e.g. Andrews 1976), was also insufficient because it could not describe the static friction data nor its connection to rate-dependent dynamic friction. This represented a major limitation not only to modeling laboratory data, but also to the problem of Figure 1 (a) Measurements of the relative variation in static friction with hold time for initially bare rock surfaces (solid symbols) and granular fault gouge (open symbols). The data have been offset to µ s = 0.6 at 1 s and thus represent relative changes in static friction. (b) Friction data versus displacement, showing measurements of static friction and µ s in slide-hold-slide experiments. Hold times are given below. In this case the loading velocity before and after holding V s/r was 3 µm/s (data from Marone 1998). (c) The relative dynamic coefficient of friction is shown versus slip velocity for initially bare rock surfaces (solid symbols) and granular fault gouge (open symbols). The data have been offset to µ d = 0.6 at 1 µm/s. (d ) Data showing the transient and steady-state effect on friction (see Figure 2 for identification of friction parameters) of a change in loading velocity for a 3-mm thick layer of quartz gouge sheared under nominally dry conditions at 25-MPa normal stress (data from J Johnson & C Marone, manuscript in preparation).

8 650 MARONE modeling repetitive stick-slip failure and the seismic cycle, which requires slip weakening to initiate unstable failure but also a healing process to reset strength between failure events. Introduction of the rate and state friction laws resolved these problems. Static friction and its observed time dependence was revealed as a special case of velocity-dependent friction, and the full range of time and displacementdependent variations illustrated in Figure 1 could be modeled with a single friction law. However, important differences exist in the laws introduced by Dieterich (1979) and Ruina (1983). Friction Evolution Laws and the State Variable Dieterich s original constitutive law (1979) stressed the importance of contact time, and thus the connection between time dependence of static friction (Figure 1a) and velocity dependence of sliding friction (Figure 1c) was via an effective contact time derived from the ratio of a critical slip distance D c to slip velocity V. Dieterich interpreted D c as representing the slip necessary to renew surface contacts; hence, the ratio D c /V defined an average contact lifetime θ. This provides a connection between time and velocity dependence of friction, which may be written (using a modern form that allows easy comparison with the other laws discussed below): ( ) V µ = µ o + aln +bln V o ( Vo θ D c ), (1) where µ o is a constant appropriate for steady-state slip at velocity V o, V is the frictional slip rate, θ is a state variable (Ruina 1983), and a and b are empirical constants. The form of Equation 1, and in particular the log terms, is suggested by the basic data of Figure 1. Dieterich s original law (1979) differed somewhat from Equation 1, in that it did not include specific reference to a state variable; however, its basic features were the same. If the state variable θ has the interpretation of a characteristic contact lifetime, then the terms in Equation 1 scaled by the constants a and b represent ratios of velocity to a reference velocity V o, and their summation describes the observed time and velocity dependence of friction (Figure 1). Of course, this statement omits many important details, but it includes the basic idea (Dieterich 1978, 1979) that restrengthening of friction during quasistationary contact can be accounted for with the same (state) memory effects and history dependence necessary to describe velocity dependence of steady-state sliding friction. To model details of friction arising from perturbations in average contact lifetime (state) or slip velocity, Equation 1 must be coupled with a description of state evolution:

9 Dieterich law ( V µ = µ o + aln V o ) +bln ( Vo θ FRICTION LAWS AND FAULTING 651 D c ), dθ dt = 1 Vθ D c. (2) Although only the second relation (the state evolution law) in Equation 2 differs between the three rate/state friction laws discussed, I include the friction relation (Equation 1) for completeness and because the full constitutive law is defined by both relations. Also, the friction relations have been extended to include variations in normal stress (Linker & Dieterich 1992; see also Wang & Scholz 1994); however, I do not consider such effects here. Equation 2 has been referred to as the slowness law and the Dieterich-Ruina law; however, for simplicity and to avoid confusion between laws, I refer to it simply as the Dieterich law. In this equation, state, and thus friction, evolve even for truly stationary contact with V = 0, which has been referred to as aging (Tullis et al 1993, Beeler et al 1994, Perrin et al 1995). It must be noted, however, that Equation 1 is undefined at V = 0. Although this condition is problematic for numerical computation, it is consistent with a definition of friction that is distinct from a generalized brittle shear strength or failure criteria. That is, by definition, friction is the normalized shear strength of a particular (existing) surface, and to be measured (including even static friction), the surface must undergo slip at some scale (e.g. Scholz et al 1972, Baumberger et al 1994). Ruina (1983) proposed a different evolution law in which velocity and slip, rather than time, were of primary importance: Ruina law ( V µ = µ o + aln V o ) +bln ( Vo θ D c ), dθ dt = Vθ D c ln ( V θ D c ), (3) where it is understood that the full constitutive law is defined by both parts of Equation 3 (the symbols have been defined above). In this view, the importance of truly stationary contact for static friction and strength recovery is discounted, and the connection between time dependence of static friction and velocity dependence of sliding friction is via an effective velocity derived from the ratio of D c to the time of quasistationary contact. Thus, while Dieterich s model casts friction primarily in terms of time dependence and static friction, Ruina s model takes the opposite view, such that any change in friction, including strengthening during quasistationary contact, requires slip. In particular, Ruina used data showing that friction exhibits memory effects, in the form of a critical slip distance required to effect a change from one value to another (Rabinowicz 1951, 1958), and precursive slip prior to unstable failure (Scholz et al 1972) to argue that all changes in friction involve slip. Time dependence of static friction, in this view, is due to slip occurring

10 652 MARONE Figure 2 Friction versus normalized displacement is shown for the three rate and state friction laws discussed. Constitutive parameters are defined at the top and apply to each curve. The effect of a step increase and decrease in load point velocity is shown for each law using the parameters given. V o = 1 µm/s. The curves were calculated for the following parameters: a = 0.01, b = 0.015, D c = 20 µm, and k = 0.01 µm 1. during the hold period, and this is not inconsistent with the observations, since the stress reduction during holding (Figure 1b) results from slip. Although the distinction between the two views of friction evolution is fundamental in terms of micromechanical interpretation of the underlying processes, the laws reproduce laboratory data in a similar fashion (Figure 2). That is, in each case a longer-term evolution process, which leads to velocity weakening in some cases (Figure 1c), competes with a shorter-term direct effect, in which friction increases for an increase in load point velocity and decreases upon a decrease in velocity (Dieterich 1979). The laws differ in their predicted responses to step increases and decreases of velocity (Figure 2). In Dieterich s law, because of the importance of effective contact time, the slip necessary for friction to regain a steady state following a perturbation scales with velocity, and thus the friction displacement curves for velocity changes of opposite sign are asymmetric (Figure 2). In contrast, the approach to a steady state is independent of time and thus symmetric with respect to velocity changes for Ruina s law. A third law has been proposed recently by Perrin, Rice, and Zheng (Perrin et al 1995). Their law exhibits both aging and symmetry with respect

11 FRICTION LAWS AND FAULTING 653 to velocity changes: PRZ law ( V µ = µ o + aln V o ) ( ) Vo θ +bln, 2D c dθ dt ( ) Vθ 2 = 1. (4) 2D c This law yields a response similar to that of the others (Figure 2), with the decay to steady state for a linearized perturbation proportional to exp( u/d c ), where u is slip. For steady-state sliding, each of the laws gives (a b) = dµ ss /dln(v), (5) and thus the slope µ d versus V on Figure 1c is given by (b a)ln(10). To model variations in frictional strength, the constitutive law must be coupled with a description of elastic interaction between the frictional surfaces and their surroundings. Tullis (1988) reviewed several key aspects of such modeling and showed the relationship between laboratory data and predictions based on stability analyses. In laboratory experiments, single-degree-of-freedom elastic coupling is generally sufficient: dµ/dt = k(v l V), (6) where V l is the velocity of a load point, inertia is taken to be negligible, and the frictional surface is assumed to be rigid so that all elastic deformation is accounted for by the spring constant k of the testing machine, here expressed as the ratio of elastic stiffness to normal stress. Which State Evolution Law? The question of which law best describes friction evolution in the laboratory and for seismogenic faults is of great interest because despite their apparent similarity (Figure 2), the different evolution laws yield qualitatively different behavior in simulations of seismic phenomena (Horowitz & Ruina 1989, Rice 1993, Perrin et al 1995, Beeler & Tullis 1996, Wennerberg & Sharp 1997). A common theme derived from these works is that friction laws that exhibit true aging are, under some conditions, required to reproduce certain features of dynamic faulting, including Gutenberg-Richter like frequency-magnitude statistics (Rice 1993) and slip-pulse rupture propagation (Heaton 1990, Perrin et al 1995, Beeler & Tullis 1996). Unfortunately, distinguishing between the laws in the laboratory, even at room temperature, has proven difficult. The symmetry of velocity changes has not provided a robust means of distinguishing between the laws. Early experiments seemed to find in favor of Ruina s law (Ruina 1983, Tullis & Weeks 1986, Marone et al 1990), and recent work on the effect of normal stress indicates that state evolution may be more closely related to slip than time (Linker & Dieterich 1992); however, the distinction is

12 654 MARONE subtle and often unresolvable owing to noise and other trends in the data. This is particularly true for studies focusing on perturbations around steady state (Tullis et al 1993). An obvious test to distinguish between the Dieterich and Ruina evolution laws would be to completely remove the shear load during holds when measuring static friction. In this case, no slip would occur during the hold period, and thus the Ruina law predicts no change in θ or µ, in contrast to predictions of an aging law. However, conducting such an experiment is difficult because the shear load must be removed and reapplied quickly, without reversal, and the surfaces must be held in position to within a fraction of D c. A few studies have been done in which shear load is partially removed and held constant during holds (e.g. Nakatani & Mochizuki 1996), and these show that the relative change in static friction, µ s = µ s µ o, scales inversely with the magnitude of the shear load reduction. However, an alternative approach has produced the most promising results to date (Beeler et al 1994). Beeler and coworkers recognized that by varying the effective stiffness of their testing machine, they could independently control the slip and time during a slide-hold-slide test (Figure 1b). They conducted experiments using the natural stiffness k n of their testing machine and an artificially higher stiffness (k s > k n ) produced by servocontrol, in which the displacement of a load point, rather than the force, is controlled, resulting in an artificially high stiffness. Their data are reproduced in Figure 3. The measurements of µ s as a function of hold time show that the healing rate (β = µ s per decade change in hold time) is independent of stiffness (Figure 3). By comparing their results with predictions of the constitutive laws, they were able to show conclusively that their data obey an aging law, such as that of Dieterich. Two aspects of Beeler et al s (1994) data and analysis are of particular interest. First, within the scatter in the data, they observed that both the absolute values of healing and the healing rate are identical for the experiments at high and low stiffness. Numerical simulations using the Dieterich law and a given loading velocity (V s/r, see Figure 1b) show that β is not a function of stiffness at long hold times, but that the absolute values of µ s vary directly with stiffness (Figure 3; Beeler et al 1994). This is consistent with the form of Equation 3, which specifies a competition between time (which causes strengthening) and slip (which reduces strength), since a smaller amount of slip occurs during a given hold for the higher stiffness. On the other hand, Ruina s law predicts that both µ s and β decrease with increasing stiffness; hence, a more compliant system would yield greater static frictional strength for a given hold time. Beeler et al (1994) focused primarily on the healing rate and showed numerical simulations of the friction law calculated for a single V s/r. However, their experiments were conducted at different velocities: V s/r was 1.0 and µm/s for k n and k s, respectively.

13 FRICTION LAWS AND FAULTING 655 Figure 3 Time dependence of the change in static friction (see Figure 1b) for initially bare granite surfaces, as measured for two stiffnesses of the testing apparatus k n = µm 1 and k s = µm 1, expressed as stiffness normalized by normal stress (data from Beeler et al 1994). Also shown are predictions of the rate and state friction laws computed using the stiffnesses and loading velocities V s/r given and the following friction constitutive values, as reported by Beeler et al (1994): a = 0.008, b = 0.009, D c = 3.0 µm. (Figure modified from Beeler et al 1994.) In Figure 3, I show numerical simulations using Beeler et al s (1994) experimental conditions and actual loading velocities for k n and k s. The comparison shows that for the Dieterich law, both the absolute values of µ s and the healing rate are the same, as indicated by their data. Thus, agreement between their data and predictions of the Dieterich law is improved by accounting for the actual velocities. On the other hand, this accounting enhances difference between the Ruina law predictions for k n and k s and thus moves these further from the experimental observations. The PRZ law shows a smaller effect of stiffness and loading velocity compared with the Ruina law; however, the predicted values of µ s and healing rate do not match the observations (Figure 3). Thus, the data of Beeler et al (1994) indicate that rock friction evolves even during truly stationary contact, at least for the conditions of their study. This is consistent with mechanistic interpretations in which the real area of surface contact increases with time, owing to interpenetration and/or creep (Dieterich 1978, Dieterich & Conrad 1984, Kato et al 1993). However, the generality

14 656 MARONE of this conclusion as applied to other conditions of temperature and chemical environment remains to be tested, as does the connection between healing measurements and the behavior observed for velocity perturbations. LABORATORY AND FIELD OBSERVATIONS OF FRICTIONAL HEALING A straightforward extension of the healing data and modeling results discussed above indicates that static friction and the rate of frictional restrengthening during quasistationary contact vary with loading rate. This has an interesting implication for the application of laboratory friction data to the problem of fault healing. In addition, it implies that the simple connection often assumed between variations in static and dynamic friction is flawed. In this section, I briefly outline these issues, drawing from recent laboratory results and field estimates of fault healing. The Rate of Frictional Healing Figure 4 shows data from experiments similar to those illustrated in Figures 1 and 3, in which static friction was measured for a range of hold times (Marone 1998). In these experiments, layers of granular quartz powder were sheared within rough granite surfaces at 25-MPa normal stress, but unlike the techniques used for Figure 3, the same apparatus stiffness was used for each set of tests. The data show that static friction and time-dependent healing µ s vary systematically with loading rate. Related effects have been demonstrated by Johnson (1981) and Kato et al (1992), as well as by the data of Beeler et al (1994), as discussed above. The data of Figure 4 have two important implications. First, they indicate that laboratory measurements of static friction and time-dependent healing must be scaled appropriately for comparison with seismic estimates of fault healing. That is, the data show that frictional healing is a system response and thus, like measurements of velocity-dependent friction, must be subject to modeling in order to recover the governing constitutive parameters (Marone 1998). The constitutive parameters may be used in applying the results to different conditions; however, without such modeling, the static friction values are strictly applicable only to the particular laboratory-testing machine with which they were measured. Second, the observation of a loading rate effect on static friction (Figure 4) indicates that the healing mechanism is not just a function of time. The observations indicate that µ s varies approximately as the product of loading rate and hold time, which implies that the mechanism of frictional strengthening is a function of slip and time. This is a feature of the rate and state friction

15 FRICTION LAWS AND FAULTING 657 Figure 4 Time dependence of static friction for loading with V s/r = 1 and 10 µm/s. The data indicate that static friction and healing vary with loading rate and therefore that static friction is a system response. Lines represent best fit log-linear relations. (Figure from Marone 1998.) laws, and indeed, when combined with a description of elastic interaction such as Equation 6, the laws show that for a given set of constitutive parameters, µ s increases with loading velocity, as is observed. However, the cause of this effect is complex. It is not simply an effect of the second term in Equation 1, the so-called direct effect, as would be implied by replacing V in this equation with V s/r. That is, slide-hold-slide tests measure relative changes in friction, and thus to compare tests conducted at different velocities, the term V o in Equation 1 must also be changed. Moreover, because of the way static friction tests are carried out, the a term of Equation 1 is zero when static friction is measured. This can be seen by noting (Figure 1b) that static friction is a local maximum and thus a point at which µ s /dt = 0, which from Equation 6 indicates that surface slip velocity equals loading velocity. However, the direct effect term is important because it moderates the amount of slip that occurs during a hold and thus influences the amount by which the frictional state changes (Marone 1998). These points have two important implications when coupled with the results of Beeler et al (1994), showing that healing exhibits Dieterich-law style aging. First, since slip velocity approaches zero in the limit of long hold time, from

16 658 MARONE Equation 2 with dθ/dt = 1 in the limit, frictional healing rate is given by bln(t). Thus the healing rate for long holds should scale as b, as noted by Beeler et al (1994). Second, the influence of loading velocity on healing rate must enter through an effect on state θ. This is confirmed by modeling, which shows that for larger initial values of V s/r, surface slip velocity decreases faster during the initial time increments of the hold, and V reaches lower values in a given time (Marone 1998). From the Dieterich evolution law, lower V results in a higher value of state, which yields larger values of µ s. Although rate and state friction laws are capable of describing complex behavior such as velocity-dependent healing, most laboratory studies of healing have considered only a single loading rate. Also, systematic measurements have tended to be restricted to static friction, without reference to other features of the data, such as the minimum friction reached at the end of the hold. Careful attention to such factors offers the possibility of providing further tests of the friction laws and possibly additional constraints on frictional behavior (e.g. Karner et al 1997). The Rate of Fault Healing Laboratory friction measurements at room temperature and for the conditions expected in the nucleation region of large earthquakes (Karner et al 1997) show that frictional healing proceeds linearly with log time during quasistationary contact. This is consistent with seismic estimates of fault healing (Kanamori & Allen 1986, Scholz et al 1986, Vidale et al 1994, Marone et al 1995), which also show an approximately log-linear strengthening rate. However, as pointed out originally by Scholz et al (1986) and Cao & Aki (1986), there is a large apparent discrepancy in the rates. Rock friction increases by only a few percent of its absolute value per decade in time (Figure 1), whereas seismic stress drop increases by a factor of 2 5 per decade increase in earthquake recurrence interval. A number of possible explanations for this discrepancy have been suggested (Scholz 1990, Marone et al 1995). For example, differences in the time scale and chemical conditions of laboratory samples and tectonic faults have been noted (Scholz 1990, Wong & Zhao 1990). Also, Scholz (1990) discussed several explanations relating to differences in the frictional properties of faults from different tectonic regimes and with different slip rates and total offsets. However, the work of Vidale et al (1994) and Marone et al (1995) on earthquakes that repeatedly rupture the same fault patch showed roughly the same healing rate as that inferred from different faults; therefore this explanation is not likely. Rather, as suggested by Marone et al (1995), the discrepancy may arise from differences in the way healing rate is measured in the lab and from seismic data.

17 FRICTION LAWS AND FAULTING 659 In laboratory studies of frictional healing, small changes are observed in the absolute value of friction as a function of hold time. The changes are but a few percent of the nominal friction value. For example, µ is 0.6 and the healing rate is 0.01 per decade change in hold time, expressed as a coefficient of friction (Figure 4). As applied to earthquake rupture and seismic stress drop, the healing data indicate that static frictional yield strength would vary by the same amount, say 1 MPa per decade for a fault under a normal stress of 100 MPa. For our nominal friction value, shear strength of this fault would be 60 MPa. Thus if earthquake stress drop σ were complete, such that σ = σ o σ f (where σ o is initial stress, which must equal the static frictional yield strength in the initiation region) and σ f is the final stress, given by dynamic frictional strength (which must be zero for complete stress drop), then the friction healing data indicate a negligible change in stress drop. For a factor of 10 increase in waiting time between events, stress drop would increase by only about 2%. On the other hand, if stress drop is a fraction of the total strength, such as expected on the basis of laboratory friction data (e.g. Scholz 1989, 1990, but see also Byerlee 1990, Hickman 1991, Rice 1992, and Beroza & Zoback 1993 for a fuller discussion of these issues), healing would result in a larger effect on stress drop. Moreover, if, instead of focusing on the percentage change in stress drop and frictional yield strength, the expected absolute changes in σ are compared, then the discrepancy essentially vanishes. That is, from the above example, the frictional healing data indicate a change of stress drop of about 1 MPa per decade increase in waiting time between events. This is actually rather close to the average values reported from scaling relations (Scholz 1990) and repeating earthquakes (Marone et al 1995). The estimate from repeating earthquakes was 1 3 MPa per decade. Of course, the value given here from laboratory data would be reduced if fault normal stress were lower. Also, the healing rate used came from room temperature experiments, whereas higher values are indicated from work using more realistic conditions (Fredrich & Evans 1992, Karner et al 1997). Thus, from the available field and laboratory observations, the discrepancy in healing rate is not great. Implications for the Relationship Between Static and Dynamic Friction Rate and state constitutive laws specify a continuum of friction values as a function of slip rate and evolving surface state. However, owing to their historical significance and everyday relevance, the terms static and dynamic friction remain in common use. As a result, simple relationships between them are often sought, and in particular the connection between time-dependent changes in static friction and velocity-dependent changes in dynamic friction has been of significant interest. A typical assumption is that variations in static friction

18 660 MARONE can be related to variations in dynamic friction by casting the time of static contact as an average velocity, using D c or another characteristic length scale (e.g. Scholz 1990). A similar assumption is made implicitly by Cao & Aki (1986) in their analysis of fault healing. However, these assumptions are incorrect and lead to misinterpretations when applied to laboratory and seismic data. From the data shown in Figure 4 and associated discussions, the variation of static friction with hold time is dµ s /dt bln(t) for long hold times. On the other hand, from the definition of the friction rate parameter, dynamic friction varies as dµ d /dlnv = a b. Variations in µ s and µ d may be compared by plotting static friction versus inverse time and dynamic friction versus velocity. Figure 5 shows such a plot using the data for quartz gouge from Figures 1b and 4. Hold time and velocity are nondimensionalized, and the data are offset for comparison so that dynamic friction at 1 µm/s = 0.6 and static friction = 0.6 for V s/r = 10 µm/s and t h = 1 s. As expected, the static and dynamic friction data show different slopes because static friction scales as b, whereas dynamic friction scales as b a (Figure 5). Moreover, the offset in static friction due to Figure 5 Static and dynamic friction are shown versus nondimensional velocity and inverse nondimensional time. The data sets are each from experiments on quartz fault gouge and indicate different slopes for static and dynamic friction, as expected from analysis of the rate and state friction laws. The inconsistency in slope indicates that variations in dynamic and static friction are not related through a simple scaling of static hold time as an average velocity, as is often assumed. Data are the same as those in Figure 1c and Figure 4.

19 FRICTION LAWS AND FAULTING 661 V s/r illustrates the problem of using such data without accounting for the effect of loading velocity. Scholz (1990) made a plot similar to Figure 5 (his figure 2.18) using friction measurements for different materials and found consistent trends across the data sets of static and dynamic friction, in contrast to the result given here. He plotted both types of data versus velocity by normalizing hold time for static friction measurements by the characteristic friction distance D c. However, the effect of this normalization is only an offset of the data, and thus this cannot explain the differences in comparison with Figure 5. Rather, it is likely that subtle differences in the friction constitutive parameters of the different data sets used, due to displacement-dependent effects or differences in materials, result in a fortuitous correlation. Such effects are discussed in the following section, although in a somewhat different context. EFFECTS OF DISPLACEMENT AND SHEAR LOCALIZATION ON CONSTITUTIVE PROPERTIES OF ROCK AND SIMULATED FAULT GOUGE A central goal of laboratory and theoretical studies of rock friction has been to identify the mechanical conditions and constitutive properties that distinguish stable from unstable sliding. In the context of rate and state friction laws, stability analyses show that this distinction is governed by the friction rate parameter a b and the critical slip distance D c (Rice 1983, Rice & Ruina 1983, Gu et al 1984), with potentially unstable sliding for a b 0 and inherently stable slip otherwise. Thus, a major preoccupation of laboratory experimentalists for the past decade or so has been to determine these friction parameters for a range of conditions, with the hope that key processes can be identified and appropriate scaling relations can be derived to connect the laboratory data with field observations. In this section, I summarize recent work in this area, with particular focus on the role of fault gouge and dilatancy and the effects of shear displacement, strain, and shear zone dimensions. The Scaling Problem Many aspects of laboratory friction experiments are highly idealized relative to natural faults, and at first glance it is not at all clear that results from laboratorysized samples can provide information relevant to earthquake faulting. However, despite significant differences in fault dimensions and other factors, it is well established that the first-order aspects of shallow earthquakes are reproduced in laboratory experiments, including observations of aseismic creep, earthquake-like instability, and the transition to dominantly stable deformation as a function of increasing temperature (e.g. Scholz 1990). Other similarities

20 662 MARONE and examples have been noted in the above discussion, and of those, the similarity between laboratory and seismic estimates of the fault healing rate may be highlighted. The explanation for this apparent utility is presumed to derive from the nondimensional nature of the coefficient of friction and perhaps from self-similarity of earthquake rupture processes with respect to scale. However, nondimensionality extends only to the strength terms that give rise to friction, whereas slip stability depends on both variations in strength and on the slip distance over which these variations occur. Thus, while it is expected that some general aspects of friction can be applied more or less directly from the laboratory to field conditions (Raleigh et al 1976, Byerlee 1978), details related to slip stability must be scaled appropriately. As applied to rate and state friction laws, this implies that the nondimensional rate parameters a and b may be used directly but that the critical slip distance must be scaled; a conjecture that is supported by the available modeling studies and field-based estimates of friction parameters (e.g. Tse & Rice 1986, Scholz 1988a, Lorenzetti & Tullis 1989, Marone et al 1991, Dieterich 1992, Power & Tullis 1992, Marone & Kilgore 1993, Rice 1993, Dieterich 1994, Dieterich & Kilgore 1996a, Tullis 1996, Ben-Zion & Rice 1997, Marone 1998). A second aspect of the scaling problem involves the effects of displacement, strain, and fault zone structure on the friction rate parameter a b. Like the issues related to D c and its scaling, the fundamental problem here involves a practical limitation. That is, laboratory experiments cannot reproduce the total displacements ( m) nor roughnesses of mature fault zones, which in turn means that factors deriving from displacement and roughness, such as wear and fault zone width, shear localization, and development of microstructures, must be studied independently. However, for understanding the application in nature of laboratory measurements of a b, the approach has been different and somewhat more contentious with regard to assessment of different results, as discussed below. Finally, a much broader view of the scaling problem may be taken, with inclusion of factors such as rock type, the presence and chemistry of fluids, and extrinsic variables such as temperature, pressure, and strain rates. However, these are for the most part accessible as laboratory control variables and hence, with proper knowledge of fault zone properties and geometry, could be studied directly. Thus I do not consider such factors here. Fault Gouge and the Second-Order Nature of Friction Rate Dependence Early laboratory friction studies (Byerlee 1967) and observations of natural and experimentally produced fault zones (Engelder et al 1975, Logan et al 1979, Rutter et al 1986, Chester & Logan 1987, Marone & Scholz 1989, Chester

21 FRICTION LAWS AND FAULTING 663 et al 1993, Beeler et al 1996) indicate that one of the most important scaling factors involves the presence and internal structure of fault gouge. Byerlee s (1967) original work showed that the accumulation of fault gouge tended to stabilize slip relative to shear between bare rock surfaces, and subsequent work, in which crushed rock and other materials were introduced to simulate fault gouge (Engelder et al 1975), confirmed this result. The connection between these two observations involves wear, as discussed by Scholz (1987) and Power et al (1988). Recently, Wong et al (1992) experimentally demonstrated the stabilizing effects of wear. These studies indicate that gouge zones widen and friction behavior evolves with accumulated displacement, which illustrates that laboratory friction results must be scaled to account for fault gouge and cumulative slip when applied in nature. However, in most cases, the effect of fault gouge and the variations in friction velocity dependence are quite subtle (Figure 6). Dieterich & Kilgore (1994), in a detailed study of the relationship between frictional behavior and surface contact properties, observed that the effect of granite fault gouge on the friction rate parameter was quite small compared to rate-dependent effects observed for other materials. Their data show that initially bare granite surfaces exhibit approximately the same steady-state friction rate dependence as granite separated by a thick (1 mm) layer of crushed granite fault gouge (for highly localized shear at the rock boundary, as discussed more fully below), with friction parameter a approximately equal to b in each case. A much greater effect is observed for D c, which is significantly larger for the case involving gouge (Figure 6). Moreover, their observations indicate similarity in the behavior of a wide range of materials, which involve different friction deformation mechanisms, and thus imply broad applicability of the rate and state friction formalism (Dieterich & Kilgore 1994, 1996a). Friction parameters are known to vary with shear displacement (Dieterich 1981), and thus differences in slip may explain part of the similarity in a b for shear with and without fault gouge, as shown in Figure 6. Nevertheless, the fact that such similarity in behavior is observed for materials sheared in the same experimental configuration indicates that it is not surprising that different investigators have found slightly dissimilar results regarding friction velocity dependence (Dieterich 1979, 1981, Johnson 1981, Solberg & Byerlee 1984, Lockner et al 1986, Morrow et al 1986, Tullis & Weeks 1986, Blanpied et al 1987, 1991, 1995, Marone & Scholz 1988, Tullis 1988, Biegel et al 1989, Morrow & Byerlee 1989, Marone et al 1990, Wong & Zhao 1990, Wong et al 1992, Kilgore et al 1993, Beeler et al 1996). These works all include experiments conducted at room temperature and with quartzo-feldspathic materials, yet they employ slightly different experimental conditions and testing configurations; some investigators observe velocity strengthening, and others velocity weakening